Subprojectivity in Abelian Categories

A new approach to projectivity in the categories of complexes

“…It is clear by Proposition 3.18 that R [1] and R both hold in Pr −1 C (R) (R), but R does not. Therefore, the subprojectivity domain of R is not closed under kernels of epimorphisms.…”

“…Let M be a complex and (N n ) n∈Z be a family of modules. Then, [1,Proposition 3.1]). Then, by Lemma 3.7 we get that for every n ∈ Z,…”

“…However, the study of the subprojectivity goes beyond that goal and, indeed, provides, among other things, a new and interesting perspective on some other known notions. In this way, an alternative perspective on the projectivity of an object of an abelian category A with enough projectives was investigated in [1] where, in addition, it was shown that subprojectivity can be used to measure characteristics different from the projectivity and that subprojectivity domains may not be restricted to a single object.…”

“…On the contrary, the subprojectivity domains of a whole class of objects can be computed, giving rise to very interesting characterizations. For instance, the subprojectivity domain of the whole class of DG-projective complexes is very useful to measure the exactness of complexes (see [1,Proposition 2.5]).…”

“…Recall that, given two objects M and N of A , M is said to be N-subprojective if for every epimorphism g : B → N and every morphism f : M → N, there exists a morphism h : M → B such that gh = f , or equivalently, if every morphism M → N factors through a projective object (see [1,Proposition 2.7]). The subprojectivity domain of any object M, denoted Pr −1 A (M), is defined as the class of all objects N such that M is N-subprojective, and the subprojectivity domain of a whole class C of A , Pr −1 A (C), is defined as the class of objects N such that every C of C is N-subprojective.…”